For the atmospheric Euler equations (), we construct
numerical fluxes \({\mathbf {\mathcal {D}}}_k\)  such that the conservative entropy
shuffle (REF ) is satisfied. Here we use a construction based on the
two-point entropy conserving flux of [1]} and
[2]}.
First, we define the following auxiliary quantities:
\({\left(\rho u_{i}\right)}^{*} &= \left\lbrace \!\left\lbrace \rho \right\rbrace \!\right\rbrace _{\log } \left\lbrace \!\left\lbrace u_{i}\right\rbrace \!\right\rbrace ,\\p^{*} &= \frac{\left\lbrace \!\left\lbrace \rho \right\rbrace \!\right\rbrace }{2\left\lbrace \!\left\lbrace b\right\rbrace \!\right\rbrace },\\e^{*} &=\frac{1}{2(\gamma - 1)\left\lbrace \!\left\lbrace b\right\rbrace \!\right\rbrace _{\log }}+ \left\lbrace \!\left\lbrace \phi (\mathbf {x})\right\rbrace \!\right\rbrace + {\Vert \left\lbrace \!\left\lbrace \mathbf {u}\right\rbrace \!\right\rbrace \Vert }_{2}^{2} - \frac{\left\lbrace \!\left\lbrace {\Vert \mathbf {u}\Vert }_{2}^{2}\right\rbrace \!\right\rbrace }{2},\\\hat{\rho }^{-} &= \frac{\left\lbrace \!\left\lbrace b\right\rbrace \!\right\rbrace \left\lbrace \!\left\lbrace \rho \right\rbrace \!\right\rbrace _{\log }}{b^-},\) 
